Consistent circle: Difference between revisions

(7 intermediate revisions by 3 users not shown)

Line 1:

Intuitively, a ~~'''consistent'''~~ [[circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).

Intuitively, a [[consistent circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).

== Motivation ==

The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there is present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)

The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there are present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) = 15 moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)

== Definitions ==

We define a ~~'''consistent'''~~ [[circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.

We define a [[consistent circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack its best approximation ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.

Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.

Line 28:

This is a much more common type of circle that is still generally reliable for most purposes, hence useful to distinguish.

When a circle satisfies this consistency but no stronger, we say its consistency is ''weak''. If it fails to satisfy this bound but generates a subset, it might qualify for a [[having a sub-weak circle]].

When a circle satisfies this consistency but no stronger, we say its consistency is ''weak''. If it fails to satisfy this bound but generates a subset, it might qualify for [[#having a sub-weak circle]].

=== Strongly consistent circle ===

Line 59:

(We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)

When a circle satisfies this strongest sense of consistency (which is only possible if the circle does not generate the full edo), we say its consistency is ''super-strong''.

== Examples ==

Line 80:

Line 82:

* [[Telicity]]

* [[Circle of fifths]]

* [[Fractional-octave temperaments]]

[[Category: pages with idiosyncratic terms]]

@@ Line 1: / Line 1: @@
-Intuitively, a '''consistent''' [[circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).
+Intuitively, a [[consistent circle]] of ~a/b's in an [[edo]] describes a case where ''a''/''b'' is so accurately approximated that you can rely on navigating with it being [[consistent]] with respect to the "circle of notes" defined. It is closely related to the concept of [[telicity]], except that a circle can be of ''any'' [[JI]] interval, as long as it is [[consistency#mathematical definition|consistently mapped]] to the unison when octave-reduced. (It also admits generalization to any [[equave]].) Note that the precise term ''consistent circle of ~a/b's'' may be shortened to ''circle of a/b's''{{idiosyncratic}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).
 == Motivation ==
-The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there is present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)
+The circle of fifths/fourths can be confusing to navigate in edos which are not [[telic]] in that a circle of [[3/2]]'s / [[4/3]]'s fails to "close", for example in [[31edo]] where the difference between 31 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there are present other intervals, such as [[5/4]] and [[7/4]] in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their [[octave complement]]s 31 times, as long as there isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form [[#Weak circle|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) = 15 moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)
 == Definitions ==
-We define a '''consistent''' [[circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack it ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.
+We define a [[consistent circle]] (abbreviatable to just ''circle''{{idiosyncratic}}) of some (usually [[JI]]) [[interval]] ''a''/''b'' as: an interval with such extremely low [[relative error]] with respect to ''N''-[[edo]] that when we stack its best approximation ''m'' > 0 times, where ''m'' is the minimum required to reach a whole number of octaves, the combined interval is [[consistent]] with its actual (untempered) size, which is to say it is off by less than 0.5\''N'' = 1200{{cent}} / ''N'' / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the edo if the circle occurs in a subset edo, but that the circle must have low enough error that within the full edo it is still consistent.
 Note that when a/b ''does'' generate all notes of the edo (meaning ''N'' = ''m''), then that means that (''a''/''b'')<sup>''m'' = ''N''</sup> reaches ''m'' = ''N'' octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.
@@ Line 28: / Line 28: @@
 This is a much more common type of circle that is still generally reliable for most purposes, hence useful to distinguish.
-When a circle satisfies this consistency but no stronger, we say its consistency is ''weak''. If it fails to satisfy this bound but generates a subset, it might qualify for a [[having a sub-weak circle]].
+When a circle satisfies this consistency but no stronger, we say its consistency is ''weak''. If it fails to satisfy this bound but generates a subset, it might qualify for [[#having a sub-weak circle]].
 === Strongly consistent circle ===
@@ Line 59: / Line 59: @@
 (We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)
+When a circle satisfies this strongest sense of consistency (which is only possible if the circle does not generate the full edo), we say its consistency is ''super-strong''.
 == Examples ==
@@ Line 80: / Line 82: @@
 * [[Telicity]]
 * [[Circle of fifths]]
+* [[Fractional-octave temperaments]]
 [[Category: pages with idiosyncratic terms]]